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Mirrors > Home > MPE Home > Th. List > 0nelopabOLD | Structured version Visualization version GIF version |
Description: Obsolete version of 0nelopab 5525 as of 3-Oct-2024. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0nelopabOLD | ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elopab 5485 | . . 3 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) | |
2 | nfopab1 5176 | . . . . . 6 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
3 | 2 | nfel2 2926 | . . . . 5 ⊢ Ⅎ𝑥∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} |
4 | 3 | nfn 1861 | . . . 4 ⊢ Ⅎ𝑥 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} |
5 | nfopab2 5177 | . . . . . . 7 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑} | |
6 | 5 | nfel2 2926 | . . . . . 6 ⊢ Ⅎ𝑦∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} |
7 | 6 | nfn 1861 | . . . . 5 ⊢ Ⅎ𝑦 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} |
8 | vex 3450 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
9 | vex 3450 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | opnzi 5432 | . . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅ |
11 | nesym 3001 | . . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ≠ ∅ ↔ ¬ ∅ = ⟨𝑥, 𝑦⟩) | |
12 | pm2.21 123 | . . . . . . . 8 ⊢ (¬ ∅ = ⟨𝑥, 𝑦⟩ → (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})) | |
13 | 11, 12 | sylbi 216 | . . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ≠ ∅ → (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})) |
14 | 10, 13 | ax-mp 5 | . . . . . 6 ⊢ (∅ = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) |
15 | 14 | adantr 482 | . . . . 5 ⊢ ((∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) |
16 | 7, 15 | exlimi 2211 | . . . 4 ⊢ (∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) |
17 | 4, 16 | exlimi 2211 | . . 3 ⊢ (∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) |
18 | 1, 17 | sylbi 216 | . 2 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) |
19 | id 22 | . 2 ⊢ (¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) | |
20 | 18, 19 | pm2.61i 182 | 1 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2944 ∅c0 4283 ⟨cop 4593 {copab 5168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-opab 5169 |
This theorem is referenced by: (None) |
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